On the Convergence of Halley’s Method for Multiple Polynomial Zeros

Abstract

In this paper, we investigate the local convergence of Halley’s method for the computation of a multiple polynomial zero with known multiplicity. We establish two local convergence theorems for Halley’s method for multiple polynomial zeros under different initial conditions. The convergence of these results is cubic right from the first iteration. Also we find an initial condition which guarantees that an initial guess is an approximate zero of the second kind for Halley’s method. All of the results are new even in the case of simple zeros.